In Differential Topology by Hirsch, I find
3.1 Theorem Let $M$ be a connected $n$-manifold and $f,g$: $D^k \hookrightarrow M$ embeddings of the $k$-disk, $0 \leq k \leq n$. If $k = n$ and $M$ is orientable, assume that $f$ and $g$ both preserve or both reserve, orientation. Then $f$ and $g$ are isotopic. If $f(D^k) \cup g(D^k) \subset M \setminus \partial M$, an isotopy between them can be realized by a diffeotopy of $M$ having compact support.
and I'm having trouble understanding "we can further assume that $f$ and $g$ both preserve or reverse orientation as embeddings into the orientable manifold $U$" in this proof.
The assumptions before the relevant section are that $\partial M = \emptyset$ and $(U, \phi)$ is a chart on $M$ such that $\phi(U) = \mathbb{R}^n$ and $\phi(f(0)) = 0$, and we have $f(D^k) \cup g(D^k) \subset U$.
The following is a quotation from the proof of the theorem.
If $M$ is orientable this follows from the hypothesis. If $M$ is not orientable we can replace $f$, if necessary, by an isotopic embedding obtained by isotoping $f$ around an orientation reversing loop based at $f(0)$.
What operations do the bolded sections represent? Thank you.
P.S.: I think the opration represents that transforming $f$ by isotopy such that isotopy that overhangs $U$. But how do I confirm that it is possible?